The most common form of imaging radar is SAR. Carl Wiley of the Goodyear Corporation, a legacy company of Lockheed Martin, invented the SAR concept in 1952. The first airborne SAR was flown on a DC-3 in 1953. Since then, SAR has become a ubiquitous imaging technique, capable of all-weather sub-meter resolution imaging.
In conventional SAR systems, the radar generates radio frequency (RF) images with near-optical quality by coherently integrating the return of a frequency-chirped radar as it travels over some distance in order to form a synthetic aperture for the antenna. Conventional SAR employs a transmitted radio frequency (RF) signal with a frequency chirp to interrogate a target region (i.e., a patch) at different angular views. Target reflections (i.e., a reflected RF signal) lead to a received signal, and the image is formed from the phase information from the received signal. Basically, the bandwidth B of a SAR resolves range, with a resolution of
                              s          range                =                  c                      2            ⁢                                                  ⁢            B                                              (        1        )            
where c is the speed of light.
In the case of a conventional “spotlight-mode” SAR, the radar stays pointed at the center of a target patch. The different views typically result from the radar traveling by the target on a path, as shown in FIG. 1, which is a schematic diagram 100 illustrating spotlight-mode SAR parameters and definitions using a plane-based SAR. A target patch on the ground is interrogated at some distance orthogonal to a flight path. The direction away from the flight path is called “down-range” and the direction on the ground parallel to the flight path is called “cross-range.” The SAR transmits at a frequency that chirps through a frequency bandwidth. Each chirp occurs very fast, and for simplicity, the plane can be considered stationary during the chirp since the distance traveled between chirps is very small. These chirps are repeated as the plane moves through discrete angular positions along the flight path relative to the target patch center, with a chirp and the associated reflected signal received at these specific angular locations.
For a conventional spotlight mode SAR, phase interferometry is used to resolve cross-range, with a cross-range resolution of
                              s                      cross            -            range                          =                              λ            ⁢                                                  ⁢            R                    S                                    (        2        )            
where λ is the wavelength of the SAR's center frequency, S is the distance along the path over which the different angular views are taken, and R is the distance from the path to the target (known as the offset). Phase interferometry requires that there is a negligible phase drift in the carrier frequency over the entire viewing path with length A, on the order of 45° or less.
In a conventional spotlight mode SAR, at each angular location, the received data is mixed with the transmission signal (i.e., a carrier) as a function for the chirped frequency, both in-phase (resulting in the in-phase signal I) and out-of-phase (resulting in the quadrature signal Q). In other words, the radar return is mixed with the in-phase and out-of-phase transmit signal, which gives I and Q. A key aspect of SAR is that the constructed quantity I+jQ is very nearly the two-dimensional Fourier transform G(kx,ky) of the target's scattering function (also known as the reflection function), defined as g(x,y), where (x=0,y=0), defines the center of the target, since the angular frequency divided by the speed of light, 2πf/c, is the wavenumber k (the Fourier transform of real space).
The quantity I+jQ can be inverted to approximately find the reflection function g(x,y) within the target patch. Importantly, this inversion can be thought of as first integrating along the direction of view (known as “de-ramping” because the Fourier integral is over the frequency chirp) and then integrating along the angular spread of the wavenumber (which, from a fundamental theory of Fourier transforms, is the same angle as the physical viewing angle). The first de-ramping integral identifies the distance to the reflectors in the target patch, but cannot resolve their relative cross-range positions.
The de-ramping process bins down-range locations into “range cells”, shown as lines along the target patch in FIG. 1. These cells are the width of the SAR's range resolution. For example, points A and B in FIG. 1 would be indistinguishable from the de-ramping process. Importantly, each return signal from each range cell has a phase of I and Q associated with the offset of that position from the target patch center. The second Fourier integral is equivalent to an integral correlating these phases of each individual reflector, which change as the angular view changes.
The resolution of the SAR image (i.e., the range cell size) in the down-range direction is 2B/c, where B is the SAR's frequency bandwidth and the resolution in the cross-range direction is the SAR's carrier wavelength divided by the total angular view on the target (λ/θ, in radians). To get fine down-range resolution, a very high bandwidth is needed. To get fine cross-range resolution, both a small carrier wavelength and a large angular view of the target patch are needed. Increasing the carrier frequency leads to increased phase drift over the viewing path and increased bandwidth leads to increased deviation of I+jQ from G(kx,ky), known as phase distortion. Combined, these effects lead to fundamental limitations for conventional spotlight SARs in achieving very fine resolution of targets that need long integration paths due to large offset distances.
Here, and in the following example, two-dimensional scattering and imaging is considered. The extension to three dimensions is straightforward and follows the conventional extension for three-dimensional SAR algorithms. The domain of the constructed quantity G(kx,ky)≈I+jQ in Fourier space is shown in graph 200 of FIG. 2.
Note that the Fourier-space wavenumbers appear through the frequency chirp, i.e., kx=(4πf/c) cos θ and ky=(4πf/c) sin θ, where f is the transmit frequency and θ is the view angle (which is the same angle in Fourier space by the Slice Projection Theorem). The Fourier transformed scattering function is
                              G          ⁡                      (                                          k                x                            ,                              k                y                                      )                          =                              1                          2              ⁢                                                          ⁢              π                                ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                g                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                  e                                      -                                          j                      ⁡                                              (                                                                              xk                            x                                                    +                                                      yk                            y                                                                          )                                                                                            ⁢                                                                  ⁢                dx                ⁢                                                                  ⁢                dy                                                                        (        3        )            
and the original scattering function can be recovered from
                              g          ⁡                      (                          x              ,              y                        )                          =                              1                          2              ⁢                                                          ⁢              π                                ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                G                  ⁡                                      (                                                                  k                        x                                            ,                                              k                        y                                                              )                                                  ⁢                                  e                                      -                                          j                      ⁡                                              (                                                                              xk                            x                                                    +                                                      yk                            y                                                                          )                                                                                            ⁢                                                                  ⁢                                  dk                  x                                ⁢                                                                  ⁢                                  dk                  y                                                                                        (        4        )            
However, the approximation of G(kx,ky) is not known over the entire k-space. Rewriting in polar coordinates to indicate the limits of the received SAR data, the reconstructed scattering image using direct Fourier inversion is then given by
                              f          ⁡                      (                          x              ,              y                        )                          =                              1                          2              ⁢                                                          ⁢              π                                ⁢                                    ∫                                                π                  /                  2                                -                                  θ                  0                                                                              π                  /                  2                                +                                  θ                  0                                                      ⁢                                          ∫                                  k                  min                                                  k                  max                                            ⁢                                                G                  ⁡                                      (                                          k                      ,                      θ                                        )                                                  ⁢                                  e                                      -                                          j                      ⁡                                              (                                                                              xkcos                            ⁢                                                                                                                  ⁢                            θ                                                    +                                                      yksin                            ⁢                                                                                                                  ⁢                            θ                                                                          )                                                                                            ⁢                                                                  ⁢                k                ⁢                                                                  ⁢                dk                ⁢                                                                  ⁢                d                ⁢                                                                  ⁢                θ                                                                        (        5        )            
where kmin and kmax are respectively given by 4π(fc−B/2)/c and 4π(fc+B/2)/c functions of the carrier frequency fc and the bandwidth B, and the angular views extend from π/2−θ0 to π/2+θ0. Note that (x cos θ+y sin θ) is the distance δx,y of (x,y) projected onto the view angle from the target center with the geometry shown in FIG. 3. The function ƒ can be rewritten as
                              f          ⁡                      (                          x              ,              y                        )                          =                              1                          2              ⁢                                                          ⁢              π                                ⁢                                    ∫                                                π                  /                  2                                -                                  θ                  0                                                                              π                  /                  2                                +                                  θ                  0                                                      ⁢                                          ∫                                  k                  min                                                  k                  max                                            ⁢                                                G                  ⁡                                      (                                          k                      ,                      θ                                        )                                                  ⁢                                  e                                                            -                      jk                                        ⁢                                                                                  ⁢                                          δ                                              x                        ,                        y                                                                                            ⁢                                                                  ⁢                k                ⁢                                                                  ⁢                dk                ⁢                                                                  ⁢                d                ⁢                                                                  ⁢                θ                                                                        (        6        )            
There are several approaches for reconstructing the scattering image, including polar reformatting and convolution backprojection (CBP) for direct Fourier inversion. Additionally, standard inversion techniques like the Maximum Entropy Method can be used to improve on the direct Fourier inversion. In polar reformatting, the data in Fourier space is first interpolated onto a Cartesian grid and then a Cartesian two-dimensional FFT is performed to recover ƒ(x,y), requiring O(N2 log N) operations for an N×N SAR image. In CBP, ƒ(x,y) is directly calculated from Eq. (6) using the polar coordinates, typically requiring O(N3) operations. Since FFTs require a Cartesian frame, non-FFT Fourier transforms need to be numerically completed for CBP. Several schemes have been proposed to reduce the complexity of the CBP approach down to O(N2 log N). Despite the typically larger computational complexity, CBP has the advantage that the image can be initially formed with few azimuthal views and continually improved as more views are added, making it especially attractive to applications where limited computer memory is available.
A significant issue with conventional spotlight SAR is that it requires phase coherency of the transmitter over the entire pass over the target. For relatively high radar frequencies, such as those needed for fine resolutions (e.g., 100 GHz is needed for approximately centimeter-scale resolutions), coherency can only be maintained for seconds. However, tens of minutes are often required to complete the angular views. Accordingly, an improved approach to SAR imaging may be beneficial.